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Free products and the lack of state-preserving approximations of nuclear $ C^*$-algebras


Author: Caleb Eckhardt
Journal: Proc. Amer. Math. Soc. 141 (2013), 2719-2727
MSC (2010): Primary 46L05, 46L09
DOI: https://doi.org/10.1090/S0002-9939-2013-11702-8
Published electronically: April 8, 2013
MathSciNet review: 3056562
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Abstract: Let $ A$ be a homogeneous $ C^*$-algebra and $ \phi $ a state on $ A.$ We show that if $ \phi $ satisfies a certain faithfulness condition, then there is a net of finite-rank, unital completely positive, $ \phi $-preserving maps on $ A$ that tend to the identity pointwise. This, combined with results of Ricard and Xu, shows that the reduced free product of homogeneous $ C^*$-algebras with respect to these states has the completely contractive approximation property. We also give an example of a faithful state on $ M_2\otimes C[0,1]$ for which no such state-preserving approximation of the identity map exists, thus answering a question of Ricard and Xu.


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Caleb Eckhardt
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47906
Email: caeckhar@math.purdue.edu

DOI: https://doi.org/10.1090/S0002-9939-2013-11702-8
Received by editor(s): November 15, 2010
Received by editor(s) in revised form: October 28, 2011
Published electronically: April 8, 2013
Additional Notes: A portion of this work was completed while the author was funded by the research program ANR-06-BLAN-0015 and by NSF grant DMS-1101144
Communicated by: Marius Junge
Article copyright: © Copyright 2013 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.